Complexities for generalized models of self-assembly

In this paper, we extend Rothemund and Winfree's examination of the tile complexity of tile self-assembly [6]. They provided a lower bound of Ω(log <i>N</i>/log log <i>N</i>) on the tile complexity of assembling an <i>N</i> × <i>N</i> square for almost all <i>N</i>. Adleman et al. [1] gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size <i>O</i>(√log <i>N</i>) which assembles an <i>N</i> × <i>N</i> square in a model which allows flexible glue strength between non-equal glues (This was independently discovered in [3]). This result is matched by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the Ω(log <i>N</i>/log log <i>N</i>) lower bound applies to <i>N</i> × <i>N</i> squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length <i>N</i> and width <i>k</i>, we provide a tighter lower bound of Ω(<i>N</i>(1/<i>k</i>)/<i>k</i>) for the standard model, yet we also give a construction which achieves <i>O</i>(log <i>N</i>/log log <i>N</i>) complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape, and show that this problem is NP-hard.